Bounds on the α-Distance Energy and α-Distance Estrada Index of Graphs

Let G be a simple undirected connected graph, then Dα(G) � αTr(G) + (1 − α)D(G) is called the α-distance matrix of G, where α ∈ [0, 1], D(G) is the distance matrix of G, and Tr(G) is the vertex transmission diagonal matrix of G. In this paper, we study some bounds on the α-distance energy and α-distance Estrada index of G. Furthermore, we establish the relation between α-distance Estrada index and α-distance energy.


α-Distance Energy of Graphs.
In this paper, we suppose that G is a connected graph. Let G � (V(G), E(G)) be a graph with the vertex set V(G) � v 1 , . . . , v n and edge set E(G). e distance between two vertices v i , v j ∈ V(G) is the length of the shortest path between v i and v j , denoted by For some properties of distance matrix, see [1][2][3].
e adjacency matrix of the graph G is A(G) � (a ij ) ∈ R n×n , where a ij � 1 if (i, j) ∈ E(G) and a ij � 0 otherwise. e Laplacian matrix and signless Laplacian matrix of G are L(G) � D(G) − A(G) and Q(G) � D(G) + A(G), respectively, where D(G) � diag(d v 1 , . . . , d v n ) ∈ R n×n and d v i is the degree of v i , i � 1, 2, . . . , n.
Graph energy has important applications in the fields of mathematics and chemistry. ere are many research studies on the above kinds of graph energy. Scholars gave the bounds on the energy of graphs, for example, the McClelland's bounds [17], Koolen-Moulton's bounds [18] and so on [19]. In [16], the distance energy of some graphs was calculated.
In [11], Guo and Zhou extended the concept of graph energy to a more general form called α-distance energy: where σ α

α-Distance Estrada Index of Graphs.
In [20], a spectral quantity is put forward by Estrada.

Main Work.
In this paper, we study some bounds on the α-distance energy and α-distance Estrada index of graphs in terms of the parameter α and the vertex number, the transmission of vertices and Wiener index. Furthermore, we establish the relation between α-distance Estrada index and α-distance energy.

Some Bounds for the α-Distance Energy of Graphs
To begin with this section, we introduce some notations and propositions.
Proposition 1 (see [6]). Let G be a graph with n vertices. en, In the following, a new matrix is established: where I n denotes identity matrix of order n. Let Proposition 2. Let G be a graph with n vertices. en, Proof. In order to prove equation (5) and (6), we have By equation (5), we have en, 2 Discrete Dynamics in Nature and Society In the following, we introduce some Lemmas which are helpful for the following proofs of theorems.
Lemma 1 (see [6]). Let G be a graph with n vertices. en, the equality holds if and only if G is a transmission regular graph.
Lemma 2 (see [27]). Let G be a graph with n vertices. en, the equality holds if and only if G � K n . K n denotes a complete graph with n vertices. Next, we give some bounds for the α-distance energy of a graph by using the parameter α and the vertex number.

Theorem 1. Let G be a connected graph with n vertices. en,
the equality holds if and only if G � K n .
By Lemma 1 and α ∈ [0, 1], we know that Suppose that ι is the largest number such that σ α (ι) (G) ≥ (2W(G)α/n). It follows from equation (5) that From Lemmas 1 and 2, we have e above three inequalities are the equality holds if and only if G � K n .

□
We give some bounds for α-distance energy through the order n, the transmission of vertex and the parameter α based on Cauchy-Schwarz inequalities in the following.

Theorem 2. Let G be a graph with n vertices. en,
Using equations (7) and (8), we have So, Similarly, from equation (11), we know According to arithmetic-geometric inequality, we have By equations (7) and (9), we have Discrete Dynamics in Nature and Society 3 In the following, we can obtain another lower bound in terms of the vertex number and the maximum value of |η α (i) (G)| of U (α) (G).
□ Corollary 1. Let G be a graph with n vertices, then By equation (7), we have

□
In the following, we obtained some new bounds for α-distance energy through the Ozeki [28] and Polya's [29] inequality, respectively. Lemma 3 (see [29]). Suppose a i and b i are real numbers for where Lemma 4 (see [28]). If a i and b i are real numbers for where M 1 � max 1≤i≤n a i , M 2 � max 1≤i≤n b i , m 1 � min 1≤i≤n a i , and m 2 � min 1≤i≤n b i .
It follows from the above Proposition the following result holds directly.

Bounds for the α-Distance Estrada Index of Graphs
In this section, some bounds for α-distance Estrada index are obtained in terms of Wiener index, the transmission of the vertex, spectral radius of D α (G), and the vertex number. Furthermore, we give the relation between α-distance Estrada index and α-distance energy. Next, we establish some bounds on the α-distance Estrada index.

□
In the following, we obtained a lower bound on the α-distance Estrada index by arithmetic-geometric inequality.

Theorem 6. Let G be a graph with n vertices. en,
From arithmetic-geometric inequality and equation (5), we obtain By means of a power-series expansion, we have □ By substituting equations (47) and (48) in equation (46), we see that Theorem 7. Let G be a graph with n vertices. en, Proof. Let f(x) � (x − 1) − ln x, where x > 0. Obviously, f(x) is a decreasing function when x ∈ (0, 1], and f(x) is increasing when x ∈ [1, +∞). en, f(x) ≥ f(1) � 0, that is, and the equality holds if and only if x � 1. So, by this function and equation (5), we have Discrete Dynamics in Nature and Society where σ α (1) (G), . . . , σ α (n) (G) are the eigenvalues of D α (G).
We are inspired by literature [32], and we give eorems 8 and 9 as follows.